Pythagorean Theorem Calculator

Calculate the hypotenuse or missing leg of a right triangle

units
units
a b c
Pythagorean Theorem Formula

a² + b² = c²

  • Where:
    • a and b are the lengths of the legs
    • c is the length of the hypotenuse
  • To find hypotenuse (c):

    c = √(a² + b²)

  • To find a leg (a or b):

    a = √(c² - b²)

Multiple Calculation Types

Calculate either the hypotenuse or a missing leg depending on what values you know.

Step-by-Step Solution

See detailed steps showing exactly how the calculation was performed.

Visual Diagram

Clear visual representation of the right triangle with labeled sides.

About the Pythagorean Theorem Calculator

Our Pythagorean Theorem Calculator is an essential tool for students, teachers, engineers, architects, and anyone working with right triangles. This powerful calculator helps you find the length of the hypotenuse or a missing leg when you know the other two sides of a right triangle.

How to Use This Calculator

  1. Select whether you want to calculate the hypotenuse or a missing leg
  2. Enter the known lengths in the appropriate fields
  3. Choose your desired decimal precision
  4. Click "Calculate" to see the result with step-by-step solution

Understanding the Pythagorean Theorem

The Pythagorean Theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (the legs). This relationship is expressed by the equation:

a² + b² = c²

Where:

  • a and b are the lengths of the legs
  • c is the length of the hypotenuse

Practical Applications

The Pythagorean Theorem has numerous real-world applications:

  • Construction: Ensuring right angles in building foundations and structures
  • Navigation: Calculating shortest distances between points
  • Engineering: Designing components with right-angle connections
  • Surveying: Measuring distances and elevations
  • Computer Graphics: Calculating distances between points in 2D and 3D space
  • Physics: Resolving vectors into components

Example Problems

Example 1: Finding the Hypotenuse

If a right triangle has legs of 3 units and 4 units, what is the length of the hypotenuse?

Solution:

c = √(3² + 4²) = √(9 + 16) = √25 = 5 units

Example 2: Finding a Missing Leg

If a right triangle has a hypotenuse of 13 units and one leg of 5 units, what is the length of the other leg?

Solution:

b = √(13² - 5²) = √(169 - 25) = √144 = 12 units